fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
IicProdIoi (a : ι) :
((Π i : Iic a, X i) × (Π i : Set.Ioi a, X i)) ≃ᵐ (Π n, X n) where
toFun := fun x i ↦ if hi : i ≤ a
then x.1 ⟨i, mem_Iic.2 hi⟩
else x.2 ⟨i, Set.mem_Ioi.2 (not_le.1 hi)⟩
invFun := fun x ↦ (fun i ↦ x i, fun i ↦ x i)
left_inv := fun x ↦ by
ext i
· simp [mem_Iic.1 i.2]
· simp [not_le.2 <| Set.mem_Ioi.1 i.2]
right_inv := fun x ↦ by simp
measurable_toFun := by
refine measurable_pi_lambda _ (fun i ↦ ?_)
by_cases hi : i ≤ a <;> simp only [Equiv.coe_fn_mk, hi, ↓reduceDIte]
· exact measurable_fst.eval
· exact measurable_snd.eval
measurable_invFun := Measurable.prodMk (measurable_restrict _) (Set.measurable_restrict _) | def | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | IicProdIoi | Gluing `Iic a` and `Ioi a` into `ℕ`, version as a measurable equivalence
on dependent functions. |
MeasurableEquiv.piSingleton (a : ℕ) : X (a + 1) ≃ᵐ Π i : Ioc a (a + 1), X i where
toFun x i := (Nat.mem_Ioc_succ.1 i.2).symm ▸ x
invFun x := x ⟨a + 1, right_mem_Ioc.2 a.lt_succ_self⟩
left_inv := fun x ↦ by simp
right_inv := fun x ↦ funext fun i ↦ by cases Nat.mem_Ioc_succ' i; rfl
measurable_toFun := by
simp_rw [eqRec_eq_cast]
refine measurable_pi_lambda _ (fun i ↦ (MeasurableEquiv.cast _ ?_).measurable)
cases Nat.mem_Ioc_succ' i; rfl
measurable_invFun := measurable_pi_apply _ | def | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | MeasurableEquiv.piSingleton | Identifying `{a + 1}` with `Ioc a (a + 1)`, as a measurable equiv on dependent functions. |
_root_.IocProdIoc_preimage {a b c : ι} (hab : a ≤ b) (hbc : b ≤ c)
(s : (i : Ioc a c) → Set (X i)) :
IocProdIoc a b c ⁻¹' (Set.univ.pi s) =
(Set.univ.pi <| restrict₂ (π := (fun n ↦ Set (X n))) (Ioc_subset_Ioc_right hbc) s) ×ˢ
(Set.univ.pi <| restrict₂ (π := (fun n ↦ Set (X n))) (Ioc_subset_Ioc_left hab) s) := by
ext x
simp only [Set.mem_preimage, Set.mem_pi, Set.mem_univ, IocProdIoc, forall_const, Subtype.forall,
mem_Ioc, Set.mem_prod, restrict₂]
refine ⟨fun h ↦ ⟨fun i ⟨hi1, hi2⟩ ↦ ?_, fun i ⟨hi1, hi2⟩ ↦ ?_⟩, fun ⟨h1, h2⟩ i ⟨hi1, hi2⟩ ↦ ?_⟩
· convert h i ⟨hi1, hi2.trans hbc⟩
rw [dif_pos hi2]
· convert h i ⟨lt_of_le_of_lt hab hi1, hi2⟩
rw [dif_neg (not_le.2 hi1)]
· split_ifs with hi3
· exact h1 i ⟨hi1, hi3⟩
· exact h2 i ⟨not_le.1 hi3, hi2⟩
variable [LocallyFiniteOrderBot ι] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | _root_.IocProdIoc_preimage | null |
_root_.IicProdIoc_preimage {a b : ι} (hab : a ≤ b) (s : (i : Iic b) → Set (X i)) :
IicProdIoc a b ⁻¹' (Set.univ.pi s) =
(Set.univ.pi <| frestrictLe₂ (π := (fun n ↦ Set (X n))) hab s) ×ˢ
(Set.univ.pi <| restrict₂ (π := (fun n ↦ Set (X n))) Ioc_subset_Iic_self s) := by
ext x
simp only [Set.mem_preimage, Set.mem_pi, Set.mem_univ, IicProdIoc_def, forall_const,
Subtype.forall, mem_Iic, Set.mem_prod, frestrictLe₂_apply, restrict₂, mem_Ioc]
refine ⟨fun h ↦ ⟨fun i hi ↦ ?_, fun i ⟨hi1, hi2⟩ ↦ ?_⟩, fun ⟨h1, h2⟩ i hi ↦ ?_⟩
· convert h i (hi.trans hab)
rw [dif_pos hi]
· convert h i hi2
rw [dif_neg (not_le.2 hi1)]
· split_ifs with hi3
· exact h1 i hi3
· exact h2 i ⟨not_le.1 hi3, hi⟩ | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | _root_.IicProdIoc_preimage | null |
noncomputable partialTraj (a b : ℕ) : Kernel (Π i : Iic a, X i) (Π i : Iic b, X i) :=
if h : b ≤ a then deterministic (frestrictLe₂ h) (measurable_frestrictLe₂ h)
else @Nat.leRec a (fun b _ ↦ Kernel (Π i : Iic a, X i) (Π i : Iic b, X i)) Kernel.id
(fun k _ κ_k ↦ ((Kernel.id ×ₖ ((κ k).map (piSingleton k))) ∘ₖ κ_k).map (IicProdIoc k (k + 1)))
b (Nat.le_of_not_ge h) | def | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj | Given a family of kernels `κ n` from `X 0 × ... × X n` to `X (n + 1)` for all `n`,
construct a kernel from `X 0 × ... × X a` to `X 0 × ... × X b` by iterating `κ`.
The idea is that the input is some trajectory up to time `a`, and the output is the distribution
of the trajectory up to time `b`. In particular if `b ≤ a`, this is just a deterministic kernel
(see `partialTraj_le`). The name `partialTraj` stands for "partial trajectory".
This kernel can be extended into a kernel with codomain `Π n, X n` via the Ionescu-Tulcea theorem. |
partialTraj_le (hba : b ≤ a) :
partialTraj κ a b = deterministic (frestrictLe₂ hba) (measurable_frestrictLe₂ _) := by
rw [partialTraj, dif_pos hba]
@[simp] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_le | If `b ≤ a`, given the trajectory up to time `a`, the trajectory up to time `b` is
deterministic and is equal to the restriction of the trajectory up to time `a`. |
partialTraj_self (a : ℕ) : partialTraj κ a a = Kernel.id := by rw [partialTraj_le le_rfl]; rfl
@[simp] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_self | null |
partialTraj_zero :
partialTraj κ a 0 = deterministic (frestrictLe₂ (zero_le a)) (measurable_frestrictLe₂ _) := by
rw [partialTraj_le (zero_le a)] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_zero | null |
partialTraj_le_def (hab : a ≤ b) : partialTraj κ a b =
@Nat.leRec a (fun b _ ↦ Kernel (Π i : Iic a, X i) (Π i : Iic b, X i)) Kernel.id
(fun k _ κ_k ↦ ((Kernel.id ×ₖ ((κ k).map (piSingleton k))) ∘ₖ κ_k).map (IicProdIoc k (k + 1)))
b hab := by
obtain rfl | hab := eq_or_lt_of_le hab
· simp
· rw [partialTraj, dif_neg (not_le.2 hab)] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_le_def | null |
partialTraj_succ_of_le (hab : a ≤ b) : partialTraj κ a (b + 1) =
((Kernel.id ×ₖ ((κ b).map (piSingleton b))) ∘ₖ partialTraj κ a b).map
(IicProdIoc b (b + 1)) := by
rw [partialTraj, dif_neg (by cutsat)]
induction b, hab using Nat.le_induction with
| base => simp
| succ k hak hk => rw [Nat.leRec_succ, ← partialTraj_le_def]; cutsat | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_succ_of_le | null |
partialTraj_succ_self (a : ℕ) :
partialTraj κ a (a + 1) =
(Kernel.id ×ₖ ((κ a).map (piSingleton a))).map (IicProdIoc a (a + 1)) := by
rw [partialTraj_succ_of_le le_rfl, partialTraj_self, comp_id] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_succ_self | null |
partialTraj_succ_eq_comp (hab : a ≤ b) :
partialTraj κ a (b + 1) = partialTraj κ b (b + 1) ∘ₖ partialTraj κ a b := by
rw [partialTraj_succ_self, ← map_comp, partialTraj_succ_of_le hab] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_succ_eq_comp | null |
partialTraj_comp_partialTraj (hab : a ≤ b) (hbc : b ≤ c) :
partialTraj κ b c ∘ₖ partialTraj κ a b = partialTraj κ a c := by
induction c, hbc using Nat.le_induction with
| base => simp
| succ k h hk => rw [partialTraj_succ_eq_comp h, comp_assoc, hk,
← partialTraj_succ_eq_comp (hab.trans h)] | theorem | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_comp_partialTraj | Given the trajectory up to time `a`, `partialTraj κ a b` gives the distribution of
the trajectory up to time `b`. Then plugging this into `partialTraj κ b c` gives
the distribution of the trajectory up to time `c`. |
private fst_prod_comp_id_prod {X Y Z : Type*} {mX : MeasurableSpace X}
{mY : MeasurableSpace Y} {mZ : MeasurableSpace Z} (κ : Kernel X Y) [IsSFiniteKernel κ]
(η : Kernel (X × Y) Z) [IsSFiniteKernel η] :
((deterministic Prod.fst measurable_fst) ×ₖ η) ∘ₖ (Kernel.id ×ₖ κ) =
Kernel.id ×ₖ (η ∘ₖ (Kernel.id ×ₖ κ)) := by
ext x s ms
simp_rw [comp_apply' _ _ _ ms, lintegral_id_prod (Kernel.measurable_coe _ ms),
deterministic_prod_apply' _ _ _ ms, id_prod_apply' _ _ ms,
comp_apply' _ _ _ (measurable_prodMk_left ms),
lintegral_id_prod (η.measurable_coe (measurable_prodMk_left ms))] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | fst_prod_comp_id_prod | This is a specific lemma used in the proof of `partialTraj_eq_prod`. It is the main rewrite step
and stating it as a separate lemma avoids using extensionality of kernels, which would generate
a lot of measurability subgoals. |
partialTraj_eq_prod [∀ n, IsSFiniteKernel (κ n)] (a b : ℕ) :
partialTraj κ a b =
(Kernel.id ×ₖ (partialTraj κ a b).map (restrict₂ Ioc_subset_Iic_self)).map
(IicProdIoc a b) := by
obtain hba | hab := le_total b a
· rw [partialTraj_le hba, IicProdIoc_le hba, map_comp_right, ← fst_eq, deterministic_map,
fst_prod, id_map]
all_goals fun_prop
induction b, hab using Nat.le_induction with
| base =>
ext1 x
rw [partialTraj_self, id_map, map_apply, prod_apply, IicProdIoc_self, ← Measure.fst,
Measure.fst_prod]
all_goals fun_prop
| succ k h hk =>
have : (IicProdIoc (X := X) k (k + 1)) ∘ (Prod.map (IicProdIoc a k) id) =
(IicProdIoc (h.trans k.le_succ) ∘ (Prod.map id (IocProdIoc a k (k + 1)))) ∘
prodAssoc := by
ext x i
simp only [IicProdIoc_def, MeasurableEquiv.IicProdIoc, MeasurableEquiv.coe_mk,
Equiv.coe_fn_mk, Function.comp_apply, Prod.map_fst, Prod.map_snd, id_eq,
Nat.succ_eq_add_one, IocProdIoc]
split_ifs <;> try rfl
omega
nth_rw 1 [← partialTraj_comp_partialTraj h k.le_succ, hk, partialTraj_succ_self, comp_map,
comap_map_comm, comap_prod, id_comap, ← id_map, map_prod_eq, ← map_comp_right, this,
map_comp_right, id_prod_eq, prodAssoc_prod, map_comp_right, ← map_prod_map, map_id,
← map_comp, map_apply_eq_iff_map_symm_apply_eq, fst_prod_comp_id_prod, ← map_comp_right,
← coe_IicProdIoc (h.trans k.le_succ), symm_comp_self, map_id,
deterministic_congr IicProdIoc_comp_restrict₂.symm, ← deterministic_comp_deterministic,
comp_deterministic_eq_comap, ← comap_prod, ← map_comp, ← comp_map, ← hk,
← partialTraj_comp_partialTraj h k.le_succ, partialTraj_succ_self, map_comp, map_comp,
← map_comp_right, ← id_map, map_prod_eq, ← map_comp_right]
· rfl
all_goals fun_prop
variable [∀ n, IsMarkovKernel (κ n)] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_eq_prod | This is a technical lemma saying that `partialTraj κ a b` consists of two independent parts, the
first one being the identity. It allows to compute integrals. |
partialTraj_succ_map_frestrictLe₂ (a b : ℕ) :
(partialTraj κ a (b + 1)).map (frestrictLe₂ b.le_succ) = partialTraj κ a b := by
obtain hab | hba := le_or_gt a b
· have := IsMarkovKernel.map (κ b) (piSingleton b).measurable
rw [partialTraj_succ_eq_comp hab, map_comp, partialTraj_succ_self, ← map_comp_right,
frestrictLe₂_comp_IicProdIoc, ← fst_eq, fst_prod, id_comp]
all_goals fun_prop
· rw [partialTraj_le (Nat.succ_le.2 hba), partialTraj_le hba.le, deterministic_map]
· rfl
· fun_prop | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_succ_map_frestrictLe₂ | null |
partialTraj_map_frestrictLe₂ (a : ℕ) (hbc : b ≤ c) :
(partialTraj κ a c).map (frestrictLe₂ hbc) = partialTraj κ a b := by
induction c, hbc using Nat.le_induction with
| base => exact map_id ..
| succ k h hk =>
rw [← hk, ← frestrictLe₂_comp_frestrictLe₂ h k.le_succ, map_comp_right,
partialTraj_succ_map_frestrictLe₂]
all_goals fun_prop | theorem | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_map_frestrictLe₂ | If we restrict the distribution of the trajectory up to time `c` to times `≤ b` we get
the trajectory up to time `b`. |
partialTraj_map_frestrictLe₂_apply (x₀ : Π i : Iic a, X i) (hbc : b ≤ c) :
(partialTraj κ a c x₀).map (frestrictLe₂ hbc) = partialTraj κ a b x₀ := by
rw [← map_apply _ (by fun_prop), partialTraj_map_frestrictLe₂] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_map_frestrictLe₂_apply | null |
partialTraj_comp_partialTraj' (c : ℕ) (hab : a ≤ b) :
partialTraj κ b c ∘ₖ partialTraj κ a b = partialTraj κ a c := by
obtain hbc | hcb := le_total b c
· rw [partialTraj_comp_partialTraj hab hbc]
· rw [partialTraj_le hcb, deterministic_comp_eq_map, partialTraj_map_frestrictLe₂] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_comp_partialTraj' | Same as `partialTraj_comp_partialTraj` but only assuming `a ≤ b`. It requires Markov kernels. |
partialTraj_comp_partialTraj'' {b c : ℕ} (hcb : c ≤ b) :
partialTraj κ b c ∘ₖ partialTraj κ a b = partialTraj κ a c := by
rw [partialTraj_le hcb, deterministic_comp_eq_map, partialTraj_map_frestrictLe₂] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | partialTraj_comp_partialTraj'' | Same as `partialTraj_comp_partialTraj` but only assuming `b ≤ c`. It requires Markov kernels. |
noncomputable lmarginalPartialTraj (a b : ℕ) (f : (Π n, X n) → ℝ≥0∞) (x₀ : Π n, X n) : ℝ≥0∞ :=
∫⁻ z : (i : Iic b) → X i, f (updateFinset x₀ _ z) ∂(partialTraj κ a b (frestrictLe a x₀)) | def | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | lmarginalPartialTraj | This function computes the integral of a function `f` against `partialTraj`,
and allows to view it as a function depending on all the variables.
This is inspired by `MeasureTheory.lmarginal`, to be able to write
`lmarginalPartialTraj κ b c (lmarginalPartialTraj κ a b f) = lmarginalPartialTraj κ a c`. |
lmarginalPartialTraj_le (hba : b ≤ a) {f : (Π n, X n) → ℝ≥0∞} (mf : Measurable f) :
lmarginalPartialTraj κ a b f = f := by
ext x₀
rw [lmarginalPartialTraj, partialTraj_le hba, Kernel.lintegral_deterministic']
· congr with i
simp [updateFinset]
· exact mf.comp measurable_updateFinset
variable {κ} | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | lmarginalPartialTraj_le | If `b ≤ a`, then integrating `f` against `partialTraj κ a b` does nothing. |
lmarginalPartialTraj_mono (a b : ℕ) {f g : (Π n, X n) → ℝ≥0∞} (hfg : f ≤ g) (x₀ : Π n, X n) :
lmarginalPartialTraj κ a b f x₀ ≤ lmarginalPartialTraj κ a b g x₀ :=
lintegral_mono fun _ ↦ hfg _ | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | lmarginalPartialTraj_mono | null |
lmarginalPartialTraj_eq_lintegral_map [∀ n, IsSFiniteKernel (κ n)] {f : (Π n, X n) → ℝ≥0∞}
(mf : Measurable f) (x₀ : Π n, X n) :
lmarginalPartialTraj κ a b f x₀ =
∫⁻ x : (Π i : Ioc a b, X i), f (updateFinset x₀ _ x)
∂(partialTraj κ a b).map (restrict₂ Ioc_subset_Iic_self) (frestrictLe a x₀) := by
nth_rw 1 [lmarginalPartialTraj, partialTraj_eq_prod, lintegral_map, lintegral_id_prod]
· congrm ∫⁻ _, f (fun i ↦ ?_) ∂_
simp only [updateFinset, mem_Iic, IicProdIoc_def,
frestrictLe_apply, mem_Ioc]
split_ifs <;> try rfl
all_goals cutsat
all_goals fun_prop | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | lmarginalPartialTraj_eq_lintegral_map | Integrating `f` against `partialTraj κ a b x` is the same as integrating only over the variables
from `x_{a+1}` to `x_b`. |
lmarginalPartialTraj_succ [∀ n, IsSFiniteKernel (κ n)] (a : ℕ)
{f : (Π n, X n) → ℝ≥0∞} (mf : Measurable f) (x₀ : Π n, X n) :
lmarginalPartialTraj κ a (a + 1) f x₀ =
∫⁻ x : X (a + 1), f (update x₀ _ x) ∂κ a (frestrictLe a x₀) := by
rw [lmarginalPartialTraj, partialTraj_succ_self, lintegral_map, lintegral_id_prod, lintegral_map]
· congrm ∫⁻ x, f (fun i ↦ ?_) ∂_
simp only [updateFinset, mem_Iic, IicProdIoc_def, frestrictLe_apply, piSingleton,
MeasurableEquiv.coe_mk, Equiv.coe_fn_mk, update]
split_ifs with h1 h2 h3 <;> try rfl
all_goals cutsat
all_goals fun_prop
@[measurability, fun_prop] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | lmarginalPartialTraj_succ | Integrating `f` against `partialTraj κ a (a + 1)` is the same as integrating against `κ a`. |
measurable_lmarginalPartialTraj (a b : ℕ) {f : (Π n, X n) → ℝ≥0∞} (hf : Measurable f) :
Measurable (lmarginalPartialTraj κ a b f) := by
unfold lmarginalPartialTraj
let g : ((i : Iic b) → X i) × (Π n, X n) → ℝ≥0∞ := fun c ↦ f (updateFinset c.2 _ c.1)
let η : Kernel (Π n, X n) (Π i : Iic b, X i) :=
(partialTraj κ a b).comap (frestrictLe a) (measurable_frestrictLe _)
change Measurable fun x₀ ↦ ∫⁻ z : (i : Iic b) → X i, g (z, x₀) ∂η x₀
refine Measurable.lintegral_kernel_prod_left' <| hf.comp ?_
simp only [updateFinset, measurable_pi_iff]
intro i
by_cases h : i ∈ Iic b <;> simp only [h, ↓reduceDIte] <;> fun_prop | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | measurable_lmarginalPartialTraj | null |
lmarginalPartialTraj_self (hab : a ≤ b) (hbc : b ≤ c)
{f : (Π n, X n) → ℝ≥0∞} (hf : Measurable f) :
lmarginalPartialTraj κ a b (lmarginalPartialTraj κ b c f) = lmarginalPartialTraj κ a c f := by
ext x₀
obtain rfl | hab := eq_or_lt_of_le hab <;> obtain rfl | hbc := eq_or_lt_of_le hbc
· rw [lmarginalPartialTraj_le κ le_rfl (measurable_lmarginalPartialTraj _ _ hf)]
· rw [lmarginalPartialTraj_le κ le_rfl (measurable_lmarginalPartialTraj _ _ hf)]
· rw [lmarginalPartialTraj_le κ le_rfl hf]
simp_rw [lmarginalPartialTraj, frestrictLe, restrict_updateFinset,
updateFinset_updateFinset_of_subset (Iic_subset_Iic.2 hbc.le)]
rw [← lintegral_comp, partialTraj_comp_partialTraj hab.le hbc.le]
fun_prop | theorem | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | lmarginalPartialTraj_self | Integrating `f` against `partialTraj κ a b` and then against `partialTraj κ b c` is the same
as integrating `f` against `partialTraj κ a c`. |
lmarginalPartialTraj_of_le [∀ n, IsMarkovKernel (κ n)] (c : ℕ) {f : (Π n, X n) → ℝ≥0∞}
(mf : Measurable f) (hf : DependsOn f (Iic a)) (hab : a ≤ b) :
lmarginalPartialTraj κ b c f = f := by
ext x
rw [lmarginalPartialTraj_eq_lintegral_map mf]
refine @lintegral_eq_const _ _ _ ?_ _ _ (ae_of_all _ fun y ↦ hf fun i hi ↦ ?_)
· refine @IsMarkovKernel.isProbabilityMeasure _ _ _ _ _ ?_ _
exact IsMarkovKernel.map _ (by fun_prop)
· simp_all only [coe_Iic, Set.mem_Iic, Function.updateFinset, mem_Ioc, dite_eq_right_iff]
cutsat | theorem | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | lmarginalPartialTraj_of_le | If `f` only depends on the variables up to rank `a` and `a ≤ b`, integrating `f` against
`partialTraj κ b c` does nothing. |
lmarginalPartialTraj_const_right [∀ n, IsMarkovKernel (κ n)] {d : ℕ} {f : (Π n, X n) → ℝ≥0∞}
(mf : Measurable f) (hf : DependsOn f (Iic a)) (hac : a ≤ c) (had : a ≤ d) :
lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f := by
wlog hcd : c ≤ d generalizing c d
· rw [this had hac (le_of_not_ge hcd)]
obtain hbc | hcb := le_total b c
· rw [← lmarginalPartialTraj_self hbc hcd mf, hf.lmarginalPartialTraj_of_le d mf hac]
· rw [hf.lmarginalPartialTraj_of_le c mf (hac.trans hcb),
hf.lmarginalPartialTraj_of_le d mf (hac.trans hcb)] | theorem | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | lmarginalPartialTraj_const_right | If `f` only depends on the variables uo to rank `a`, integrating beyond rank `a` is the same
as integrating up to rank `a`. |
dependsOn_lmarginalPartialTraj [∀ n, IsSFiniteKernel (κ n)] (a : ℕ) {f : (Π n, X n) → ℝ≥0∞}
(hf : DependsOn f (Iic b)) (mf : Measurable f) :
DependsOn (lmarginalPartialTraj κ a b f) (Iic a) := by
intro x y hxy
obtain hba | hab := le_total b a
· rw [Kernel.lmarginalPartialTraj_le κ hba mf]
exact hf fun i hi ↦ hxy i (Iic_subset_Iic.2 hba hi)
rw [lmarginalPartialTraj_eq_lintegral_map mf, lmarginalPartialTraj_eq_lintegral_map mf]
congrm ∫⁻ z : _, ?_ ∂(partialTraj κ a b).map _ (fun i ↦ ?_)
· exact hxy i.1 i.2
· refine hf.updateFinset _ ?_
rwa [← coe_sdiff, Iic_diff_Ioc_self_of_le hab] | theorem | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict",
"Mathlib.Probability.Kernel.Composition.Prod",
"Mathlib.Probability.Kernel.IonescuTulcea.Maps"
] | Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.lean | dependsOn_lmarginalPartialTraj | If `f` only depends on variables up to rank `b`, its integral from `a` to `b` only depends on
variables up to rank `a`. |
private Iic_pi_eq {a b : ℕ} (h : a = b) :
(Π i : Iic a, X i) = (Π i : Iic b, X i) := by cases h; rfl | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | Iic_pi_eq | null |
private cast_pi {s t : Set ℕ} (h : s = t) (x : (i : s) → X i) (i : t) :
cast (congrArg (fun u : Set ℕ ↦ (Π i : u, X i)) h) x i = x ⟨i.1, h.symm ▸ i.2⟩ := by
cases h; rfl
variable [∀ n, MeasurableSpace (X n)] | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | cast_pi | null |
private measure_cast {a b : ℕ} (h : a = b) (μ : (n : ℕ) → Measure (Π i : Iic n, X i)) :
(μ a).map (cast (Iic_pi_eq h)) = μ b := by
cases h
exact Measure.map_id | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | measure_cast | null |
private heq_measurableSpace_Iic_pi {a b : ℕ} (h : a = b) :
(inferInstance : MeasurableSpace (Π i : Iic a, X i)) ≍
(inferInstance : MeasurableSpace (Π i : Iic b, X i)) := by cases h; rfl | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | heq_measurableSpace_Iic_pi | null |
iterateInduction {a : ℕ} (x : Π i : Iic a, X i)
(ind : (n : ℕ) → (Π i : Iic n, X i) → X (n + 1)) : Π n, X n
| 0 => x ⟨0, mem_Iic.2 <| zero_le a⟩
| k + 1 => if h : k + 1 ≤ a
then x ⟨k + 1, mem_Iic.2 h⟩
else ind k (fun i ↦ iterateInduction x ind i)
decreasing_by exact Nat.lt_succ.2 (mem_Iic.1 i.2) | def | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | iterateInduction | This function takes as input a tuple `(x_₀, ..., x_ₐ)` and `ind` a function which
given `(y_₀, ...,y_ₙ)` outputs `x_{n+1} : X (n + 1)`, and it builds an element of `Π n, X n`
by starting with `(x_₀, ..., x_ₐ)` and then iterating `ind`. |
frestrictLe_iterateInduction {a : ℕ} (x : Π i : Iic a, X i)
(ind : (n : ℕ) → (Π i : Iic n, X i) → X (n + 1)) :
frestrictLe a (iterateInduction x ind) = x := by
ext i
simp only [frestrictLe_apply]
obtain ⟨(zero | j), hj⟩ := i <;> rw [iterateInduction]
rw [dif_pos (mem_Iic.1 hj)] | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | frestrictLe_iterateInduction | null |
isProjectiveLimit_nat_iff' {μ : (I : Finset ℕ) → Measure (Π i : I, X i)}
(hμ : IsProjectiveMeasureFamily μ) (ν : Measure (Π n, X n)) (a : ℕ) :
IsProjectiveLimit ν μ ↔ ∀ ⦃n⦄, a ≤ n → ν.map (frestrictLe n) = μ (Iic n) := by
refine ⟨fun h n _ ↦ h (Iic n), fun h I ↦ ?_⟩
have := (I.subset_Iic_sup_id.trans (Iic_subset_Iic.2 (le_max_left (I.sup id) a)))
rw [← restrict₂_comp_restrict this, ← Measure.map_map, ← frestrictLe, h (le_max_right _ _), ← hμ]
all_goals fun_prop | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | isProjectiveLimit_nat_iff' | To check that a measure `ν` is the projective limit of a projective family of measures indexed
by `Finset ℕ`, it is enough to check on intervals of the form `Iic n`, where `n` is larger than
a given integer. |
isProjectiveLimit_nat_iff {μ : (I : Finset ℕ) → Measure (Π i : I, X i)}
(hμ : IsProjectiveMeasureFamily μ) (ν : Measure (Π n, X n)) :
IsProjectiveLimit ν μ ↔ ∀ n, ν.map (frestrictLe n) = μ (Iic n) := by
rw [isProjectiveLimit_nat_iff' hμ _ 0]
simp
variable (μ : (n : ℕ) → Measure (Π i : Iic n, X i)) | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | isProjectiveLimit_nat_iff | To check that a measure `ν` is the projective limit of a projective family of measures indexed
by `Finset ℕ`, it is enough to check on intervals of the form `Iic n`. |
noncomputable inducedFamily (S : Finset ℕ) : Measure ((k : S) → X k) :=
(μ (S.sup id)).map (restrict₂ S.subset_Iic_sup_id) | def | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | inducedFamily | Given a family of measures `μ : (n : ℕ) → Measure (Π i : Iic n, X i)`, we can define a family
of measures indexed by `Finset ℕ` by projecting the measures. |
inducedFamily_Iic (n : ℕ) : inducedFamily μ (Iic n) = μ n := by
rw [inducedFamily, ← measure_cast (sup_Iic n) μ]
congr with x i
rw [restrict₂, cast_pi (by rw [sup_Iic n])] | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | inducedFamily_Iic | Given a family of measures `μ : (n : ℕ) → Measure (Π i : Iic n, X i)`, the induced family
equals `μ` over the intervals `Iic n`. |
isProjectiveMeasureFamily_inducedFamily
(h : ∀ a b : ℕ, ∀ hab : a ≤ b, (μ b).map (frestrictLe₂ hab) = μ a) :
IsProjectiveMeasureFamily (inducedFamily μ) := by
intro I J hJI
have sls : J.sup id ≤ I.sup id := sup_mono hJI
simp only [inducedFamily]
rw [Measure.map_map, restrict₂_comp_restrict₂,
← restrict₂_comp_restrict₂ J.subset_Iic_sup_id (Iic_subset_Iic.2 sls), ← Measure.map_map,
← frestrictLe₂.eq_def sls, h (J.sup id) (I.sup id) sls]
all_goals fun_prop | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | isProjectiveMeasureFamily_inducedFamily | Given a family of measures `μ : (n : ℕ) → Measure (Π i : Iic n, X i)`, the induced family
will be projective only if `μ` is projective, in the sense that if `a ≤ b`, then projecting
`μ b` gives `μ a`. |
isProjectiveMeasureFamily_partialTraj {a : ℕ} (x₀ : Π i : Iic a, X i) :
IsProjectiveMeasureFamily (inducedFamily (fun b ↦ partialTraj κ a b x₀)) :=
isProjectiveMeasureFamily_inducedFamily _
(fun _ _ ↦ partialTraj_map_frestrictLe₂_apply (κ := κ) x₀) | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | isProjectiveMeasureFamily_partialTraj | null |
noncomputable trajContent {a : ℕ} (x₀ : Π i : Iic a, X i) :
AddContent (measurableCylinders X) :=
projectiveFamilyContent (isProjectiveMeasureFamily_partialTraj κ x₀)
variable {κ} | def | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | trajContent | Given a family of kernels `κ : (n : ℕ) → Kernel (Π i : Iic n, X i) (X (n + 1))`, and the
trajectory up to time `a` we can construct an additive content over cylinders. It corresponds
to composing the kernels, starting at time `a + 1`. |
trajContent_cylinder {a b : ℕ} {S : Set (Π i : Iic b, X i)} (mS : MeasurableSet S)
(x₀ : Π i : Iic a, X i) :
trajContent κ x₀ (cylinder (Iic b) S) = partialTraj κ a b x₀ S := by
rw [trajContent, projectiveFamilyContent_cylinder _ mS, inducedFamily_Iic] | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | trajContent_cylinder | The `trajContent κ x₀` of a cylinder indexed by first coordinates is given by `partialTraj`. |
trajContent_eq_lmarginalPartialTraj {b : ℕ} {S : Set (Π i : Iic b, X i)}
(mS : MeasurableSet S) (x₀ : Π n, X n) (a : ℕ) :
trajContent κ (frestrictLe a x₀) (cylinder (Iic b) S) =
lmarginalPartialTraj κ a b ((cylinder (Iic b) S).indicator 1) x₀ := by
rw [trajContent_cylinder mS, ← lintegral_indicator_one mS, lmarginalPartialTraj]
congr with x
apply Set.indicator_const_eq_indicator_const
rw [mem_cylinder]
congrm (fun i ↦ ?_) ∈ S
simp [updateFinset, i.2] | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | trajContent_eq_lmarginalPartialTraj | The `trajContent` of a cylinder is equal to the integral of its indicator function against
`partialTraj`. |
trajContent_ne_top {a : ℕ} {x : Π i : Iic a, X i} {s : Set (Π n, X n)} :
trajContent κ x s ≠ ∞ :=
projectiveFamilyContent_ne_top (isProjectiveMeasureFamily_partialTraj κ x) | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | trajContent_ne_top | null |
le_lmarginalPartialTraj_succ {f : ℕ → (Π n, X n) → ℝ≥0∞} {a : ℕ → ℕ}
(hcte : ∀ n, DependsOn (f n) (Iic (a n))) (mf : ∀ n, Measurable (f n))
{bound : ℝ≥0∞} (fin_bound : bound ≠ ∞) (le_bound : ∀ n x, f n x ≤ bound) {k : ℕ}
(anti : ∀ x, Antitone (fun n ↦ lmarginalPartialTraj κ (k + 1) (a n) (f n) x))
{l : (Π n, X n) → ℝ≥0∞}
(htendsto : ∀ x, Tendsto (fun n ↦ lmarginalPartialTraj κ (k + 1) (a n) (f n) x) atTop (𝓝 (l x)))
(ε : ℝ≥0∞) (y : Π i : Iic k, X i)
(hpos : ∀ x n, ε ≤ lmarginalPartialTraj κ k (a n) (f n) (updateFinset x (Iic k) y)) :
∃ z, ∀ x n,
ε ≤ lmarginalPartialTraj κ (k + 1) (a n) (f n)
(update (updateFinset x (Iic k) y) (k + 1) z) := by
have _ n : Nonempty (X n) := by
induction n using Nat.case_strong_induction_on with
| hz => exact ⟨y ⟨0, mem_Iic.2 (zero_le _)⟩⟩
| hi m hm =>
have : Nonempty (Π i : Iic m, X i) :=
⟨fun i ↦ @Classical.ofNonempty _ (hm i.1 (mem_Iic.1 i.2))⟩
exact ProbabilityMeasure.nonempty ⟨κ m Classical.ofNonempty, inferInstance⟩
let F n : (Π n, X n) → ℝ≥0∞ := lmarginalPartialTraj κ (k + 1) (a n) (f n)
have tendstoF x : Tendsto (F · x) atTop (𝓝 (l x)) := htendsto x
have f_eq x n : lmarginalPartialTraj κ k (a n) (f n) x =
lmarginalPartialTraj κ k (k + 1) (F n) x := by
simp_rw [F]
obtain h | h | h := lt_trichotomy (k + 1) (a n)
· rw [← lmarginalPartialTraj_self k.le_succ h.le (mf n)]
· rw [← h, lmarginalPartialTraj_le _ le_rfl (mf n)]
· rw [lmarginalPartialTraj_le _ _ (mf n), (hcte n).lmarginalPartialTraj_of_le _ (mf n),
(hcte n).lmarginalPartialTraj_of_le _ (mf n)]
all_goals cutsat
have F_le n x : F n x ≤ bound := by
simpa [F, lmarginalPartialTraj] using lintegral_le_const (ae_of_all _ fun z ↦ le_bound _ _)
have tendsto_int x : Tendsto (fun n ↦ lmarginalPartialTraj κ k (a n) (f n) x) atTop
(𝓝 (lmarginalPartialTraj κ k (k + 1) l x)) := by
simp_rw [f_eq, lmarginalPartialTraj]
exact tendsto_lintegral_of_dominated_convergence (fun _ ↦ bound)
(fun n ↦ (measurable_lmarginalPartialTraj _ _ (mf n)).comp measurable_updateFinset)
(fun n ↦ Eventually.of_forall <| fun y ↦ F_le n _)
(by simp [fin_bound]) (Eventually.of_forall (fun _ ↦ tendstoF _))
have ε_le_lint x : ε ≤ lmarginalPartialTraj κ k (k + 1) l (updateFinset x _ y) :=
ge_of_tendsto (tendsto_int _) (by simp [hpos])
let x_ : Π n, X n := Classical.ofNonempty
obtain ⟨x, hx⟩ : ∃ x, ε ≤ l (update (updateFinset x_ _ y) (k + 1) x) := by
have : ∫⁻ x, l (update (updateFinset x_ _ y) (k + 1) x) ∂(κ k y) ≠ ∞ :=
ne_top_of_le_ne_top fin_bound <| lintegral_le_const <| ae_of_all _
fun y ↦ le_of_tendsto' (tendstoF _) <| fun _ ↦ F_le _ _
obtain ⟨x, hx⟩ := exists_lintegral_le this
refine ⟨x, (ε_le_lint x_).trans ?_⟩
rwa [lmarginalPartialTraj_succ, frestrictLe_updateFinset]
exact ENNReal.measurable_of_tendsto (by fun_prop) (tendsto_pi_nhds.2 htendsto)
refine ⟨x, fun x' n ↦ ?_⟩
have := le_trans hx ((anti _).le_of_tendsto (tendstoF _) n)
... | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | le_lmarginalPartialTraj_succ | This is an auxiliary result for `trajContent_tendsto_zero`. Consider `f` a sequence of bounded
measurable functions such that `f n` depends only on the first coordinates up to `a n`.
Assume that when integrating `f n` against `partialTraj (k + 1) (a n)`, one gets a non-increasing
sequence of functions which converges to `l`.
Assume then that there exists `ε` and `y : Π i : Iic k, X i` such that
when integrating `f n` against `partialTraj k (a n) y`, you get something at least
`ε` for all `n`. Then there exists `z` such that this remains true when integrating
`f` against `partialTraj (k + 1) (a n) (update y (k + 1) z)`. |
trajContent_tendsto_zero {A : ℕ → Set (Π n, X n)}
(A_mem : ∀ n, A n ∈ measurableCylinders X) (A_anti : Antitone A) (A_inter : ⋂ n, A n = ∅)
{p : ℕ} (x₀ : Π i : Iic p, X i) :
Tendsto (fun n ↦ trajContent κ x₀ (A n)) atTop (𝓝 0) := by
have _ n : Nonempty (X n) := by
induction n using Nat.case_strong_induction_on with
| hz => exact ⟨x₀ ⟨0, mem_Iic.2 (zero_le _)⟩⟩
| hi m hm =>
have : Nonempty (Π i : Iic m, X i) :=
⟨fun i ↦ @Classical.ofNonempty _ (hm i.1 (mem_Iic.1 i.2))⟩
exact ProbabilityMeasure.nonempty ⟨κ m Classical.ofNonempty, inferInstance⟩
have A_cyl n : ∃ a S, MeasurableSet S ∧ A n = cylinder (Iic a) S := by
simpa [measurableCylinders_nat] using A_mem n
choose a S mS A_eq using A_cyl
let χ n := (A n).indicator (1 : (Π n, X n) → ℝ≥0∞)
have mχ n : Measurable (χ n) := by
simp_rw [χ, A_eq]
exact (measurable_indicator_const_iff 1).2 <| (mS n).cylinder
have χ_dep n : DependsOn (χ n) (Iic (a n)) := by
simp_rw [χ, A_eq]
exact dependsOn_cylinder_indicator_const ..
have lma_const x y n :
lmarginalPartialTraj κ p (a n) (χ n) (updateFinset x _ x₀) =
lmarginalPartialTraj κ p (a n) (χ n) (updateFinset y _ x₀) := by
refine (χ_dep n).dependsOn_lmarginalPartialTraj p (mχ n) fun i hi ↦ ?_
rw [mem_coe, mem_Iic] at hi
simp [updateFinset, hi]
have χ_anti : Antitone χ := fun m n hmn y ↦ by
apply Set.indicator_le fun a ha ↦ ?_
simp [χ, A_anti hmn ha]
have lma_inv k M n (h : a n ≤ M) :
lmarginalPartialTraj κ k M (χ n) = lmarginalPartialTraj κ k (a n) (χ n) :=
(χ_dep n).lmarginalPartialTraj_const_right (mχ n) h le_rfl
have anti_lma k x : Antitone fun n ↦ lmarginalPartialTraj κ k (a n) (χ n) x := by
intro m n hmn
simp only
rw [← lma_inv k ((a n).max (a m)) n (le_max_left _ _),
← lma_inv k ((a n).max (a m)) m (le_max_right _ _)]
exact lmarginalPartialTraj_mono _ _ (χ_anti hmn) _
have this k x : ∃ l, Tendsto (fun n ↦ lmarginalPartialTraj κ k (a n) (χ n) x) atTop (𝓝 l) := by
obtain h | h := tendsto_of_antitone (anti_lma k x)
· rw [OrderBot.atBot_eq] at h
exact ⟨0, h.mono_right <| pure_le_nhds 0⟩
· exact h
choose l hl using this
have l_const x y : l p (updateFinset x _ x₀) = l p (updateFinset y _ x₀) := by
have := hl p (updateFinset x _ x₀)
simp_rw [lma_const x y] at this
exact tendsto_nhds_unique this (hl p _)
obtain ⟨ε, hε⟩ : ∃ ε, ∀ x, l p (updateFinset x _ x₀) = ε :=
⟨l p (updateFinset Classical.ofNonempty _ x₀), fun x ↦ l_const _ _⟩
... | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | trajContent_tendsto_zero | This is the key theorem to prove the existence of the `traj`:
the `trajContent` of a decreasing sequence of cylinders with empty intersection
converges to `0`.
This implies the `σ`-additivity of `trajContent`
(see `addContent_iUnion_eq_sum_of_tendsto_zero`),
which allows to extend it to the `σ`-algebra by Carathéodory's theorem. |
isSigmaSubadditive_trajContent {a : ℕ} (x₀ : Π i : Iic a, X i) :
(trajContent κ x₀).IsSigmaSubadditive := by
refine isSigmaSubadditive_of_addContent_iUnion_eq_tsum
isSetRing_measurableCylinders (fun f hf hf_Union hf' ↦ ?_)
refine addContent_iUnion_eq_sum_of_tendsto_zero isSetRing_measurableCylinders
(trajContent κ x₀) (fun _ _ ↦ trajContent_ne_top) ?_ hf hf_Union hf'
exact fun s hs anti_s inter_s ↦ trajContent_tendsto_zero hs anti_s inter_s x₀ | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | isSigmaSubadditive_trajContent | The `trajContent` is sigma-subadditive. |
noncomputable trajFun (a : ℕ) (x₀ : Π i : Iic a, X i) : Measure (Π n, X n) :=
(trajContent κ x₀).measure isSetSemiring_measurableCylinders generateFrom_measurableCylinders.ge
(isSigmaSubadditive_trajContent κ x₀) | def | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | trajFun | This function is the kernel given by the Ionescu-Tulcea theorem. It is shown below that it
is measurable and turned into a true kernel in `Kernel.traj`. |
isProbabilityMeasure_trajFun (a : ℕ) (x₀ : Π i : Iic a, X i) :
IsProbabilityMeasure (trajFun κ a x₀) where
measure_univ := by
rw [← cylinder_univ (Iic 0), trajFun, AddContent.measure_eq, trajContent_cylinder .univ,
measure_univ]
· exact generateFrom_measurableCylinders.symm
· exact cylinder_mem_measurableCylinders _ _ .univ | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | isProbabilityMeasure_trajFun | null |
isProjectiveLimit_trajFun (a : ℕ) (x₀ : Π i : Iic a, X i) :
IsProjectiveLimit (trajFun κ a x₀) (inducedFamily (fun n ↦ partialTraj κ a n x₀)) := by
refine isProjectiveLimit_nat_iff (isProjectiveMeasureFamily_partialTraj κ x₀) _ |>.2 fun n ↦ ?_
ext s ms
rw [Measure.map_apply (measurable_frestrictLe n) ms, trajFun, AddContent.measure_eq, trajContent,
projectiveFamilyContent_congr _ (frestrictLe n ⁻¹' s) rfl ms]
· exact generateFrom_measurableCylinders.symm
· exact cylinder_mem_measurableCylinders _ _ ms
variable {κ} in | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | isProjectiveLimit_trajFun | null |
measurable_trajFun (a : ℕ) : Measurable (trajFun κ a) := by
apply Measure.measurable_of_measurable_coe
refine MeasurableSpace.induction_on_inter
(C := fun t ht ↦ Measurable (fun x₀ ↦ trajFun κ a x₀ t))
(s := measurableCylinders X) generateFrom_measurableCylinders.symm
isPiSystem_measurableCylinders (by simp) (fun t ht ↦ ?cylinder) (fun t mt ht ↦ ?compl)
(fun f disf mf hf ↦ ?union)
· obtain ⟨N, S, mS, t_eq⟩ : ∃ N S, MeasurableSet S ∧ t = cylinder (Iic N) S := by
simpa [measurableCylinders_nat] using ht
simp_rw [trajFun, AddContent.measure_eq _ _ generateFrom_measurableCylinders.symm _ ht,
trajContent, projectiveFamilyContent_congr _ t t_eq mS, inducedFamily]
refine Measure.measurable_measure.1 ?_ _ mS
exact (Measure.measurable_map _ (measurable_restrict₂ _)).comp (measurable _)
· have := isProbabilityMeasure_trajFun κ a
simpa [measure_compl mt (measure_ne_top _ _)] using Measurable.const_sub ht _
· simpa [measure_iUnion disf mf] using Measurable.ennreal_tsum hf | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | measurable_trajFun | null |
noncomputable traj (a : ℕ) : Kernel (Π i : Iic a, X i) (Π n, X n) where
toFun := trajFun κ a
measurable' := measurable_trajFun a | def | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | traj | *Ionescu-Tulcea Theorem* : Given a family of kernels `κ n` taking variables in `Iic n` with
value in `X (n + 1)`, the kernel `traj κ a` takes a variable `x` depending on the
variables `i ≤ a` and associates to it a kernel on trajectories depending on all variables,
where the entries with index `≤ a` are those of `x`, and then one follows iteratively the
kernels `κ a`, then `κ (a + 1)`, and so on.
The fact that such a kernel exists on infinite trajectories is not obvious, and is the content of
the Ionescu-Tulcea theorem. |
traj_apply (a : ℕ) (x : Π i : Iic a, X i) : traj κ a x = trajFun κ a x := rfl | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | traj_apply | null |
traj_map_frestrictLe (a b : ℕ) : (traj κ a).map (frestrictLe b) = partialTraj κ a b := by
ext x
rw [map_apply, traj_apply, frestrictLe, isProjectiveLimit_trajFun, inducedFamily_Iic]
fun_prop | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | traj_map_frestrictLe | null |
traj_map_frestrictLe_apply (a b : ℕ) (x : Π i : Iic a, X i) :
(traj κ a x).map (frestrictLe b) = partialTraj κ a b x := by
rw [← map_apply _ (measurable_frestrictLe b), traj_map_frestrictLe] | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | traj_map_frestrictLe_apply | null |
traj_map_frestrictLe_of_le {a b : ℕ} (hab : a ≤ b) :
(traj κ b).map (frestrictLe a) =
deterministic (frestrictLe₂ hab) (measurable_frestrictLe₂ _) := by
rw [traj_map_frestrictLe, partialTraj_le]
variable (κ) | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | traj_map_frestrictLe_of_le | null |
eq_traj' {a : ℕ} (n : ℕ) (η : Kernel (Π i : Iic a, X i) (Π n, X n))
(hη : ∀ b ≥ n, η.map (frestrictLe b) = partialTraj κ a b) : η = traj κ a := by
ext x : 1
refine ((isProjectiveLimit_trajFun _ _ _).unique ?_).symm
rw [isProjectiveLimit_nat_iff' _ _ n]
· intro k hk
rw [inducedFamily_Iic, ← map_apply _ (measurable_frestrictLe k), hη k hk]
· exact isProjectiveMeasureFamily_partialTraj κ x | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | eq_traj' | To check that `η = traj κ a` it is enough to show that the restriction of `η` to variables `≤ b`
is `partialTraj κ a b` for any `b ≥ n`. |
eq_traj {a : ℕ} (η : Kernel (Π i : Iic a, X i) (Π n, X n))
(hη : ∀ b, η.map (frestrictLe b) = partialTraj κ a b) : η = traj κ a :=
eq_traj' κ 0 η fun b _ ↦ hη b
variable {κ} | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | eq_traj | To check that `η = traj κ a` it is enough to show that the restriction of `η` to variables `≤ b`
is `partialTraj κ a b`. |
traj_comp_partialTraj {a b : ℕ} (hab : a ≤ b) :
(traj κ b) ∘ₖ (partialTraj κ a b) = traj κ a := by
refine eq_traj _ _ fun n ↦ ?_
rw [map_comp, traj_map_frestrictLe, partialTraj_comp_partialTraj' _ hab] | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | traj_comp_partialTraj | Given the distribution up to tome `a`, `partialTraj κ a b` gives the distribution
of the trajectory up to time `b`, and composing this with `traj κ b` gives the distribution
of the whole trajectory. |
traj_eq_prod (a : ℕ) :
traj κ a = (Kernel.id ×ₖ (traj κ a).map (Set.Ioi a).restrict).map (IicProdIoi a) := by
refine (eq_traj' _ (a + 1) _ fun b hb ↦ ?_).symm
rw [← map_comp_right]
conv_lhs => enter [2]; change (IicProdIoc a b) ∘
(Prod.map id (fun x i ↦ x ⟨i.1, Set.mem_Ioi.2 (mem_Ioc.1 i.2).1⟩))
· rw [map_comp_right, ← map_prod_map, ← map_comp_right]
· conv_lhs => enter [1, 2, 2]; change (Ioc a b).restrict
rw [← restrict₂_comp_restrict Ioc_subset_Iic_self, ← frestrictLe, map_comp_right,
traj_map_frestrictLe, map_id, ← partialTraj_eq_prod]
all_goals fun_prop
all_goals fun_prop
all_goals fun_prop | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | traj_eq_prod | This theorem shows that `traj κ n` is, up to an equivalence, the product of
a deterministic kernel with another kernel. This is an intermediate result to compute integrals
with respect to this kernel. |
traj_map_updateFinset {n : ℕ} (x : Π i : Iic n, X i) :
(traj κ n x).map (updateFinset · (Iic n) x) = traj κ n x := by
nth_rw 2 [traj_eq_prod]
have : (updateFinset · _ x) = IicProdIoi n ∘ (Prod.mk x) ∘ (Set.Ioi n).restrict := by
ext; simp [IicProdIoi, updateFinset]
rw [this, ← Function.comp_assoc, ← Measure.map_map, ← Measure.map_map, map_apply, prod_apply,
map_apply, id_apply, Measure.dirac_prod]
all_goals fun_prop | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | traj_map_updateFinset | null |
lintegral_traj₀ {a : ℕ} (x₀ : Π i : Iic a, X i) {f : (Π n, X n) → ℝ≥0∞}
(mf : AEMeasurable f (traj κ a x₀)) :
∫⁻ x, f x ∂traj κ a x₀ = ∫⁻ x, f (updateFinset x (Iic a) x₀) ∂traj κ a x₀ := by
nth_rw 1 [← traj_map_updateFinset, MeasureTheory.lintegral_map']
· convert mf
exact traj_map_updateFinset x₀
· exact measurable_updateFinset_left.aemeasurable | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | lintegral_traj₀ | null |
lintegral_traj {a : ℕ} (x₀ : Π i : Iic a, X i) {f : (Π n, X n) → ℝ≥0∞}
(mf : Measurable f) :
∫⁻ x, f x ∂traj κ a x₀ = ∫⁻ x, f (updateFinset x (Iic a) x₀) ∂traj κ a x₀ :=
lintegral_traj₀ x₀ mf.aemeasurable
variable {E : Type*} [NormedAddCommGroup E] | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | lintegral_traj | null |
integrable_traj {a b : ℕ} (hab : a ≤ b) {f : (Π n, X n) → E}
(x₀ : Π i : Iic a, X i) (i_f : Integrable f (traj κ a x₀)) :
∀ᵐ x ∂traj κ a x₀, Integrable f (traj κ b (frestrictLe b x)) := by
rw [← traj_comp_partialTraj hab, integrable_comp_iff] at i_f
· apply ae_of_ae_map (p := fun x ↦ Integrable f (traj κ b x))
· fun_prop
· convert i_f.1
rw [← traj_map_frestrictLe, Kernel.map_apply _ (measurable_frestrictLe _)]
· exact i_f.aestronglyMeasurable | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | integrable_traj | null |
aestronglyMeasurable_traj {a b : ℕ} (hab : a ≤ b) {f : (Π n, X n) → E}
{x₀ : Π i : Iic a, X i} (hf : AEStronglyMeasurable f (traj κ a x₀)) :
∀ᵐ x ∂partialTraj κ a b x₀, AEStronglyMeasurable f (traj κ b x) := by
rw [← traj_comp_partialTraj hab] at hf
exact hf.comp
variable [NormedSpace ℝ E] | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | aestronglyMeasurable_traj | null |
integral_traj {a : ℕ} (x₀ : Π i : Iic a, X i) {f : (Π n, X n) → E}
(mf : AEStronglyMeasurable f (traj κ a x₀)) :
∫ x, f x ∂traj κ a x₀ = ∫ x, f (updateFinset x (Iic a) x₀) ∂traj κ a x₀ := by
nth_rw 1 [← traj_map_updateFinset, integral_map]
· exact measurable_updateFinset_left.aemeasurable
· convert mf
rw [traj_map_updateFinset] | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | integral_traj | When computing `∫ x, f x ∂traj κ n x₀`, because the trajectory up to time `n` is
determined by `x₀` we can replace `x` by `updateFinset x (Iic a) x₀`. |
partialTraj_compProd_traj {a b : ℕ} (hab : a ≤ b) (u : Π i : Iic a, X i) :
(partialTraj κ a b u) ⊗ₘ (traj κ b) = (traj κ a u).map (fun x ↦ (frestrictLe b x, x)) := by
ext s ms
rw [Measure.map_apply, Measure.compProd_apply, ← traj_comp_partialTraj hab, comp_apply']
· congr 1 with x
rw [← traj_map_updateFinset, Measure.map_apply, Measure.map_apply]
· congr 1 with y
simp only [Set.mem_preimage]
congrm (fun i ↦ ?_, fun i ↦ ?_) ∈ s <;> simp [updateFinset]
any_goals fun_prop
all_goals exact ms.preimage (by fun_prop)
any_goals exact ms.preimage (by fun_prop)
fun_prop | lemma | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | partialTraj_compProd_traj | null |
integral_traj_partialTraj' {a b : ℕ} (hab : a ≤ b) {x₀ : Π i : Iic a, X i}
{f : (Π i : Iic b, X i) → (Π n : ℕ, X n) → E}
(hf : Integrable f.uncurry ((partialTraj κ a b x₀) ⊗ₘ (traj κ b))) :
∫ x, ∫ y, f x y ∂traj κ b x ∂partialTraj κ a b x₀ =
∫ x, f (frestrictLe b x) x ∂traj κ a x₀ := by
have hf' := hf
rw [partialTraj_compProd_traj hab] at hf'
simp_rw [← uncurry_apply_pair f, ← Measure.integral_compProd hf,
partialTraj_compProd_traj hab, integral_map (by fun_prop) hf'.1] | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | integral_traj_partialTraj' | null |
integral_traj_partialTraj {a b : ℕ} (hab : a ≤ b) {x₀ : Π i : Iic a, X i}
{f : (Π n : ℕ, X n) → E} (hf : Integrable f (traj κ a x₀)) :
∫ x, ∫ y, f y ∂traj κ b x ∂partialTraj κ a b x₀ = ∫ x, f x ∂traj κ a x₀ := by
apply integral_traj_partialTraj' hab
rw [← traj_comp_partialTraj hab, comp_apply, ← Measure.snd_compProd] at hf
exact hf.comp_measurable measurable_snd | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | integral_traj_partialTraj | null |
setIntegral_traj_partialTraj' {a b : ℕ} (hab : a ≤ b) {u : (Π i : Iic a, X i)}
{f : (Π i : Iic b, X i) → (Π n : ℕ, X n) → E}
(hf : Integrable f.uncurry ((partialTraj κ a b u) ⊗ₘ (traj κ b)))
{A : Set (Π i : Iic b, X i)} (hA : MeasurableSet A) :
∫ x in A, ∫ y, f x y ∂traj κ b x ∂partialTraj κ a b u =
∫ y in frestrictLe b ⁻¹' A, f (frestrictLe b y) y ∂traj κ a u := by
rw [← integral_integral_indicator _ _ _ hA, integral_traj_partialTraj' hab]
· simp_rw [← Set.indicator_comp_right, ← integral_indicator (measurable_frestrictLe b hA)]
rfl
convert hf.indicator (hA.prod .univ)
ext ⟨x, y⟩
by_cases hx : x ∈ A <;> simp [uncurry_def, hx] | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | setIntegral_traj_partialTraj' | null |
setIntegral_traj_partialTraj {a b : ℕ} (hab : a ≤ b) {x₀ : (Π i : Iic a, X i)}
{f : (Π n : ℕ, X n) → E} (hf : Integrable f (traj κ a x₀))
{A : Set (Π i : Iic b, X i)} (hA : MeasurableSet A) :
∫ x in A, ∫ y, f y ∂traj κ b x ∂partialTraj κ a b x₀ =
∫ y in frestrictLe b ⁻¹' A, f y ∂traj κ a x₀ := by
refine setIntegral_traj_partialTraj' hab ?_ hA
rw [← traj_comp_partialTraj hab, comp_apply, ← Measure.snd_compProd] at hf
exact hf.comp_measurable measurable_snd
variable [CompleteSpace E]
open Filtration | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | setIntegral_traj_partialTraj | null |
condExp_traj {a b : ℕ} (hab : a ≤ b) {x₀ : Π i : Iic a, X i}
{f : (Π n, X n) → E} (i_f : Integrable f (traj κ a x₀)) :
(traj κ a x₀)[f|piLE b] =ᵐ[traj κ a x₀]
fun x ↦ ∫ y, f y ∂traj κ b (frestrictLe b x) := by
have i_f' : Integrable (fun x ↦ ∫ y, f y ∂(traj κ b) x)
(((traj κ a) x₀).map (frestrictLe b)) := by
rw [← map_apply _ (measurable_frestrictLe _), traj_map_frestrictLe _ _]
rw [← traj_comp_partialTraj hab] at i_f
exact i_f.integral_comp
refine ae_eq_condExp_of_forall_setIntegral_eq (piLE.le _) i_f
(fun s _ _ ↦ i_f'.comp_aemeasurable (measurable_frestrictLe b).aemeasurable |>.integrableOn)
?_ ?_ |>.symm <;> rw [piLE_eq_comap_frestrictLe]
· rintro - ⟨t, mt, rfl⟩ -
simp_rw [Function.comp_apply]
rw [← setIntegral_map mt i_f'.1, ← map_apply, traj_map_frestrictLe,
setIntegral_traj_partialTraj hab i_f mt]
all_goals fun_prop
· exact (i_f'.1.comp_ae_measurable' (measurable_frestrictLe b).aemeasurable) | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | condExp_traj | null |
condExp_traj' {a b c : ℕ} (hab : a ≤ b) (hbc : b ≤ c)
(x₀ : Π i : Iic a, X i) (f : (Π n, X n) → E) :
(traj κ a x₀)[f|piLE b] =ᵐ[traj κ a x₀]
fun x ↦ ∫ y, ((traj κ a x₀)[f|piLE c]) (updateFinset x (Iic c) y)
∂partialTraj κ b c (frestrictLe b x) := by
have i_cf : Integrable ((traj κ a x₀)[f|piLE c]) (traj κ a x₀) :=
integrable_condExp
have mcf : StronglyMeasurable ((traj κ a x₀)[f|piLE c]) :=
stronglyMeasurable_condExp.mono (piLE.le c)
filter_upwards [piLE.condExp_condExp f hbc, condExp_traj hab i_cf] with x h1 h2
rw [← h1, h2, ← traj_map_frestrictLe, Kernel.map_apply, integral_map]
· congr with y
apply stronglyMeasurable_condExp.dependsOn_of_piLE
simp only [Set.mem_Iic, updateFinset, mem_Iic, frestrictLe_apply, dite_eq_ite]
exact fun i hi ↦ (if_pos hi).symm
any_goals fun_prop
exact (mcf.comp_measurable measurable_updateFinset).aestronglyMeasurable | theorem | Probability | [
"Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent",
"Mathlib.MeasureTheory.Function.FactorsThrough",
"Mathlib.MeasureTheory.Measure.ProbabilityMeasure",
"Mathlib.MeasureTheory.OuterMeasure.OfAddContent",
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Ione... | Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean | condExp_traj' | null |
coinvariantsKer_eq_range (hg : ∀ x, x ∈ Subgroup.zpowers g) :
Coinvariants.ker ρ = LinearMap.range (ρ g - LinearMap.id) := by
refine le_antisymm (Submodule.span_le.2 ?_) ?_
· rintro a ⟨⟨γ, α⟩, rfl⟩
rcases mem_powers_iff_mem_zpowers.2 (hg γ) with ⟨i, rfl⟩
induction i with | zero => exact ⟨0, by simp⟩ | succ n _ =>
use (Fin.partialSum (fun (j : Fin (n + 1)) => ρ (g ^ (j : ℕ)) α) (Fin.last _))
simpa using ρ.apply_sub_id_partialSum_eq _ _ _
· rintro x ⟨y, rfl⟩
simpa using Coinvariants.sub_mem_ker g y | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | coinvariantsKer_eq_range | null |
noncomputable coinvariantsEquiv (hg : ∀ x, x ∈ Subgroup.zpowers g) :
ρ.Coinvariants ≃ₗ[k] (_ ⧸ LinearMap.range (ρ g - LinearMap.id)) :=
Submodule.quotEquivOfEq _ _ (coinvariantsKer_eq_range ρ g hg) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | coinvariantsEquiv | Given a finite cyclic group `G` generated by `g` and a `G` representation `(V, ρ)`, `V_G` is
isomorphic to `V ⧸ Im(ρ(g - 1))`. |
coinvariantsKer_leftRegular_eq_ker :
Coinvariants.ker (Representation.leftRegular k G) =
LinearMap.ker (linearCombination k (fun _ => (1 : k))) := by
refine le_antisymm (Submodule.span_le.2 ?_) fun x hx => ?_
· rintro x ⟨⟨g, y⟩, rfl⟩
simpa [linearCombination, sub_eq_zero, sum_fintype]
using Finset.sum_bijective _ (Group.mulLeft_bijective g⁻¹) (by aesop) (by aesop)
· have : x = x.sum (fun g r => single g r - single 1 r) := by
ext g
by_cases hg : g = 1
· simp_all [linearCombination, sum_apply']
· simp_all [sum_apply']
rw [this]
exact Submodule.finsuppSum_mem _ _ _ _ fun g _ =>
Coinvariants.mem_ker_of_eq g (single 1 (x g)) _ (by simp) | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | coinvariantsKer_leftRegular_eq_ker | null |
range_norm_eq_ker_applyAsHom_sub (hg : ∀ x, x ∈ Subgroup.zpowers g) :
LinearMap.range (leftRegular k G).norm.hom.hom =
LinearMap.ker (applyAsHom (leftRegular k G) g - 𝟙 _).hom.hom :=
le_antisymm (fun _ ⟨_, h⟩ => by simp_all [← h]) fun x hx => ⟨single 1 (x g), by
ext j; simpa [Representation.norm] using (apply_eq_of_leftRegular_eq_of_generator g hg _
(by simpa [sub_eq_zero] using hx) j).symm⟩ | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | range_norm_eq_ker_applyAsHom_sub | null |
range_applyAsHom_sub_eq_ker_linearCombination (hg : ∀ x, x ∈ Subgroup.zpowers g) :
LinearMap.range (applyAsHom (leftRegular k G) g - 𝟙 _).hom.hom =
LinearMap.ker (linearCombination k (fun _ => (1 : k))) := by
simp [← FiniteCyclicGroup.coinvariantsKer_eq_range _ _ hg,
← FiniteCyclicGroup.coinvariantsKer_leftRegular_eq_ker] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | range_applyAsHom_sub_eq_ker_linearCombination | null |
range_applyAsHom_sub_eq_ker_norm (hg : ∀ x, x ∈ Subgroup.zpowers g) :
LinearMap.range (applyAsHom (leftRegular k G) g - 𝟙 _).hom.hom =
LinearMap.ker (leftRegular k G).norm.hom.hom := by
simp [ker_leftRegular_norm_eq, ← range_applyAsHom_sub_eq_ker_linearCombination k g hg] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | range_applyAsHom_sub_eq_ker_norm | null |
@[simps]
noncomputable chainComplexFunctor : Rep k G ⥤ ChainComplex (Rep k G) ℕ where
obj A := HomologicalComplex.alternatingConst A (φ := A.norm) (ψ := applyAsHom A g - 𝟙 A)
(by ext; simp) (by ext; simp) fun _ _ => ComplexShape.down_nat_odd_add
map f := {
f i := f
comm' := by
rintro i j ⟨rfl⟩
by_cases hj : Even (j + 1)
· simp [if_pos hj, norm_comm]
· simp [if_neg hj, applyAsHom_comm] }
map_id _ := rfl
map_comp _ _ := rfl
variable {k} | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | chainComplexFunctor | Given a finite group `G` and `g : G`, this is the functor `Rep k G ⥤ ChainComplex (Rep k G) ℕ`
sending `A : Rep k G` to the periodic chain complex in `Rep k G` given by
`... ⟶ A --N--> A --(ρ(g) - 𝟙)--> A --N--> A --(ρ(g) - 𝟙)--> A ⟶ 0`
where `N` is the norm map. When `G` is generated by `g` and `A` is the left regular representation
`k[G]`, it is a projective resolution of `k` as a trivial representation.
It sends a morphism `f : A ⟶ B` to the chain morphism defined by `f` in every degree. |
noncomputable normHomCompSub : ShortComplex (ModuleCat k) :=
ShortComplex.mk A.norm.hom (applyAsHom A g - 𝟙 A).hom (by ext; simp) | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | normHomCompSub | Given a finite cyclic group `G` generated by `g : G` and a `k`-linear `G`-representation `A`,
this is the short complex in `ModuleCat k` given by `A --N--> A --(ρ(g) - 𝟙)--> A`
where `N` is the norm map. Its homology is `Hⁱ(G, A)` for even `i` and `Hᵢ(G, A)` for odd `i`. |
noncomputable subCompNormHom : ShortComplex (ModuleCat k) :=
ShortComplex.mk (applyAsHom A g - 𝟙 A).hom A.norm.hom (by ext; simp) | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | subCompNormHom | Given a finite cyclic group `G` generated by `g : G` and a `k`-linear `G`-representation `A`,
this is the short complex in `ModuleCat k` given by `A --N--> A --(ρ(g) - 𝟙)--> A`
where `N` is the norm map. Its homology is `Hⁱ(G, A)` for even `i` and `Hᵢ(G, A)` for odd `i`. |
noncomputable moduleCatChainComplex : ChainComplex (ModuleCat k) ℕ :=
HomologicalComplex.alternatingConst A.V (φ := A.norm.hom) (ψ := (applyAsHom A g - 𝟙 A).hom)
(by ext; simp) (by ext; simp) fun _ _ => ComplexShape.down_nat_odd_add | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | moduleCatChainComplex | Given a finite cyclic group `G` generated by `g : G` and a `k`-linear `G`-representation `A`,
this is the periodic chain complex in `ModuleCat k` given by
`... ⟶ A --N--> A --(ρ(g) - 𝟙)--> A --N--> A --(ρ(g) - 𝟙)--> A ⟶ 0` where `N` is the norm map.
Its homology is the group homology of `A`. |
noncomputable moduleCatCochainComplex : CochainComplex (ModuleCat k) ℕ :=
HomologicalComplex.alternatingConst A.V (φ := (applyAsHom A g - 𝟙 A).hom) (ψ := A.norm.hom)
(by ext; simp) (by ext; simp) fun _ _ => ComplexShape.up_nat_odd_add
variable (k) | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | moduleCatCochainComplex | Given a finite cyclic group `G` generated by `g : G` and a `k`-linear `G`-representation `A`,
this is the periodic chain complex in `Rep k G` given by
`0 ⟶ A --(ρ(g) - 𝟙)--> A --N--> A --(ρ(g) - 𝟙)--> A --N--> A ⟶ ...` where `N` is the norm map.
Its cohomology is the group cohomology of `A`. |
@[simps!]
noncomputable resolution.π (g : G) :
(chainComplexFunctor k g).obj (leftRegular k G) ⟶
(ChainComplex.single₀ (Rep k G)).obj (trivial k G k) :=
(((chainComplexFunctor k g).obj (leftRegular k G)).toSingle₀Equiv _).symm
⟨leftRegularHom _ 1, (leftRegularHomEquiv _).injective <| by simp [leftRegularHomEquiv]⟩ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | resolution.π | Given a finite cyclic group `G` generated by `g : G`, let `P` denote the periodic chain complex
of `k`-linear `G`-representations given by
`... ⟶ k[G] --N--> k[G] --(ρ(g) - 𝟙)--> k[G] --N--> k[G] --(ρ(g) - 𝟙)--> k[G] ⟶ 0` where `ρ` is
the left regular representation and `N` is the norm map. This is the chain morphism from `P` to
the chain complex concentrated at 0 by the trivial representation `k` used to show `P` is a
projective resolution of `k`. It sends `x : k[G]` to the sum of its coefficients. |
resolution_quasiIso (g : G) (hg : ∀ x, x ∈ Subgroup.zpowers g) :
QuasiIso (resolution.π k g) where
quasiIsoAt m := by
induction m with
| zero =>
simp only [resolution.π]
rw [ChainComplex.quasiIsoAt₀_iff, ShortComplex.quasiIso_iff_of_zeros' _ rfl rfl rfl]
constructor
· apply (Action.forget (ModuleCat k) _).reflects_exact_of_faithful
simpa [ShortComplex.moduleCat_exact_iff_range_eq_ker,
HomologicalComplex.alternatingConst, ChainComplex.toSingle₀Equiv] using
leftRegular.range_applyAsHom_sub_eq_ker_linearCombination k g hg
· rw [Rep.epi_iff_surjective]
intro x
use single 1 x
simp [ChainComplex.toSingle₀Equiv]
| succ m _ =>
rw [quasiIsoAt_iff_exactAt' (hL := ChainComplex.exactAt_succ_single_obj ..),
HomologicalComplex.exactAt_iff' _ (m + 2) (m + 1) m (by simp) (by simp)]
apply (Action.forget (ModuleCat k) _).reflects_exact_of_faithful
rw [ShortComplex.moduleCat_exact_iff_range_eq_ker]
by_cases hm : Odd (m + 1)
· simpa [if_pos (Nat.even_add_one.2 (Nat.not_even_iff_odd.2 hm)),
if_neg (Nat.not_even_iff_odd.2 hm)]
using leftRegular.range_norm_eq_ker_applyAsHom_sub k g hg
· simpa [ShortComplex.moduleCat_exact_iff_range_eq_ker, if_pos (Nat.not_odd_iff_even.1 hm),
if_neg (Nat.not_even_iff_odd.2 <| Nat.odd_add_one.2 hm)]
using leftRegular.range_applyAsHom_sub_eq_ker_norm k g hg | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | resolution_quasiIso | null |
@[simps]
noncomputable resolution (g : G) (hg : ∀ x, x ∈ Subgroup.zpowers g) :
ProjectiveResolution (trivial k G k) where
complex := (FiniteCyclicGroup.chainComplexFunctor k g).obj (leftRegular k G)
projective _ := inferInstanceAs <| Projective (leftRegular k G)
π := FiniteCyclicGroup.resolution.π k g
quasiIso := resolution_quasiIso k g hg | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.AlternatingConst",
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Homological/FiniteCyclic.lean | resolution | Given a finite cyclic group `G` generated by `g : G`, this is the projective resolution of `k`
as a trivial `k`-linear `G`-representation given by periodic complex
`... ⟶ k[G] --N--> k[G] --(ρ(g) - 𝟙)--> k[G] --N--> k[G] --(ρ(g) - 𝟙)--> k[G] ⟶ 0` where `ρ` is
the left regular representation and `N` is the norm map. |
@[deprecated "We now favour `Representation.finsuppLEquivFreeAsModule`" (since := "2025-06-04")]
ofMulActionBasisAux :
MonoidAlgebra k G ⊗[k] ((Fin n → G) →₀ k) ≃ₗ[MonoidAlgebra k G]
(ofMulAction k G (Fin (n + 1) → G)).asModule :=
haveI e := (Rep.equivalenceModuleMonoidAlgebra.1.mapIso
(Rep.diagonalSuccIsoTensorTrivial k G n).symm).toLinearEquiv
{ e with
map_smul' := fun r x => by
rw [RingHom.id_apply, LinearEquiv.toFun_eq_coe, ← LinearEquiv.map_smul e]
congr 1
refine x.induction_on ?_ (fun x y => ?_) fun y z hy hz => ?_
· simp only [smul_zero]
· rw [TensorProduct.smul_tmul', smul_eq_mul, ← ofMulAction_self_smul_eq_mul]
exact (smul_tprod_one_asModule (Representation.ofMulAction k G G) r x y).symm
· rw [smul_add, hz, hy, smul_add] } | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | ofMulActionBasisAux | The `k[G]`-linear isomorphism `k[G] ⊗ₖ k[Gⁿ] ≃ k[Gⁿ⁺¹]`, where the `k[G]`-module structure on
the left-hand side is `TensorProduct.leftModule`, whilst that of the right-hand side comes from
`Representation.asModule`. Allows us to use `Algebra.TensorProduct.basis` to get a `k[G]`-basis
of the right-hand side. |
@[simps obj map]
classifyingSpaceUniversalCover [Monoid G] :
SimplicialObject (Action (Type u) G) where
obj n := Action.ofMulAction G (Fin (n.unop.len + 1) → G)
map f :=
{ hom := fun x => x ∘ f.unop.toOrderHom
comm := fun _ => rfl }
map_id _ := rfl
map_comp _ _ := rfl | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | classifyingSpaceUniversalCover | A `k[G]`-basis of `k[Gⁿ⁺¹]`, coming from the `k[G]`-linear isomorphism
`k[G] ⊗ₖ k[Gⁿ] ≃ k[Gⁿ⁺¹].` -/
@[deprecated "We now favour `Representation.freeAsModuleBasis`; the old definition can be derived
from this and `Rep.diagonalSuccIsoFree" (since := "2025-06-05")]
alias ofMulActionBasis := Representation.freeAsModuleBasis
@[deprecated "We now favour `Representation.free_asModule_free`; the old theorem can be derived
from this and `Rep.diagonalSuccIsoFree" (since := "2025-06-05")]
alias ofMulAction_free := Representation.free_asModule_free
end Basis
end groupCohomology.resolution
variable (G)
/-- The simplicial `G`-set sending `[n]` to `Gⁿ⁺¹` equipped with the diagonal action of `G`. |
cechNerveTerminalFromIso :
cechNerveTerminalFrom (Action.ofMulAction G G) ≅ classifyingSpaceUniversalCover G :=
NatIso.ofComponents (fun _ => limit.isoLimitCone (Action.ofMulActionLimitCone _ _)) fun f => by
refine IsLimit.hom_ext (Action.ofMulActionLimitCone.{u, 0} G fun _ => G).2 fun j => ?_
dsimp only [cechNerveTerminalFrom, Pi.lift]
rw [Category.assoc, limit.isoLimitCone_hom_π, limit.lift_π, Category.assoc]
exact (limit.isoLimitCone_hom_π _ _).symm | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | cechNerveTerminalFromIso | When the category is `G`-Set, `cechNerveTerminalFrom` of `G` with the left regular action is
isomorphic to `EG`, the universal cover of the classifying space of `G` as a simplicial `G`-set. |
cechNerveTerminalFromIsoCompForget :
cechNerveTerminalFrom G ≅ classifyingSpaceUniversalCover G ⋙ forget _ :=
NatIso.ofComponents (fun _ => Types.productIso _) fun _ =>
Matrix.ext fun _ _ => Types.Limit.lift_π_apply (Discrete.functor fun _ ↦ G) _ _ _
variable (k)
open AlgebraicTopology SimplicialObject.Augmented SimplicialObject CategoryTheory.Arrow | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | cechNerveTerminalFromIsoCompForget | As a simplicial set, `cechNerveTerminalFrom` of a monoid `G` is isomorphic to the universal
cover of the classifying space of `G` as a simplicial set. |
compForgetAugmented : SimplicialObject.Augmented (Type u) :=
SimplicialObject.augment (classifyingSpaceUniversalCover G ⋙ forget _) (terminal _)
(terminal.from _) fun _ _ _ => Subsingleton.elim _ _ | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | compForgetAugmented | The universal cover of the classifying space of `G` as a simplicial set, augmented by the map
from `Fin 1 → G` to the terminal object in `Type u`. |
extraDegeneracyAugmentedCechNerve :
ExtraDegeneracy (Arrow.mk <| terminal.from G).augmentedCechNerve :=
AugmentedCechNerve.extraDegeneracy (Arrow.mk <| terminal.from G)
⟨fun _ => (1 : G),
@Subsingleton.elim _ (@Unique.instSubsingleton _ (Limits.uniqueToTerminal _)) _ _⟩ | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | extraDegeneracyAugmentedCechNerve | The augmented Čech nerve of the map from `Fin 1 → G` to the terminal object in `Type u` has an
extra degeneracy. |
extraDegeneracyCompForgetAugmented : ExtraDegeneracy (compForgetAugmented G) := by
refine
ExtraDegeneracy.ofIso (?_ : (Arrow.mk <| terminal.from G).augmentedCechNerve ≅ _)
(extraDegeneracyAugmentedCechNerve G)
exact
Comma.isoMk (CechNerveTerminalFrom.iso G ≪≫ cechNerveTerminalFromIsoCompForget G)
(Iso.refl _) (by ext : 1; exact IsTerminal.hom_ext terminalIsTerminal _ _) | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | extraDegeneracyCompForgetAugmented | The universal cover of the classifying space of `G` as a simplicial set, augmented by the map
from `Fin 1 → G` to the terminal object in `Type u`, has an extra degeneracy. |
compForgetAugmented.toModule : SimplicialObject.Augmented (ModuleCat.{u} k) :=
((SimplicialObject.Augmented.whiskering _ _).obj (ModuleCat.free k)).obj (compForgetAugmented G) | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | compForgetAugmented.toModule | The free functor `Type u ⥤ ModuleCat.{u} k` applied to the universal cover of the classifying
space of `G` as a simplicial set, augmented by the map from `Fin 1 → G` to the terminal object
in `Type u`. |
extraDegeneracyCompForgetAugmentedToModule :
ExtraDegeneracy (compForgetAugmented.toModule k G) :=
ExtraDegeneracy.map (extraDegeneracyCompForgetAugmented G) (ModuleCat.free k) | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | extraDegeneracyCompForgetAugmentedToModule | If we augment the universal cover of the classifying space of `G` as a simplicial set by the
map from `Fin 1 → G` to the terminal object in `Type u`, then apply the free functor
`Type u ⥤ ModuleCat.{u} k`, the resulting augmented simplicial `k`-module has an extra
degeneracy. |
Rep.standardComplex [Monoid G] :=
(AlgebraicTopology.alternatingFaceMapComplex (Rep k G)).obj
(classifyingSpaceUniversalCover G ⋙ linearization k G)
@[deprecated (since := "2025-06-06")]
alias groupCohomology.resolution := Rep.standardComplex | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | Rep.standardComplex | The standard resolution of `k` as a trivial representation, defined as the alternating
face map complex of a simplicial `k`-linear `G`-representation. |
d (G : Type u) (n : ℕ) : ((Fin (n + 1) → G) →₀ k) →ₗ[k] (Fin n → G) →₀ k :=
Finsupp.lift ((Fin n → G) →₀ k) k (Fin (n + 1) → G) fun g =>
(@Finset.univ (Fin (n + 1)) _).sum fun p =>
Finsupp.single (g ∘ p.succAbove) ((-1 : k) ^ (p : ℕ))
variable {k G}
@[simp] | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | d | The `k`-linear map underlying the differential in the standard resolution of `k` as a trivial
`k`-linear `G`-representation. It sends `(g₀, ..., gₙ) ↦ ∑ (-1)ⁱ • (g₀, ..., ĝᵢ, ..., gₙ)`. |
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